Back to Problems

Newton second law for the motion of an electron through a ring of charge

H.Tahsiri

```Off[General::spell];
Off[General::spell1];
Pagewidth->70;
Clear["Global`*"];
Clear[y,V,a,t]```

1) The electric field near y=0 is

```Ep = k Q y/R^3

k Q y
-----
3
R```

2) Use Newton second law to write the equation of motion

`F = ma;a=dV/dt`

3) The force and the field are related by

`F=q Ep =-e Ep`

4) The angular frequency and the period are related by

`w0=2Pi f=2Pi/T= k Q e/(m R^3)=1`

5 ) The equation of motion is written as

```equ= D[y[t],{t,2}]+ y[t]

y[t] + y''[t]```

6 ) The Solution is

```sol=DSolve[{equ==0,y[0]==y0,y'[0]==0},y[t],t][[1,1]]

y[t] -> y0 Cos[t]```

7 ) The displacement of the electron as a function of time is

```y[t_]=y0 Cos[t]

y0 Cos[t]```

8 ) The velocity of the electron as a function of time is

```V[t_]=D[y[t],t]

-(y0 Sin[t])```

9 ) The acceleration of the electron as a function of time is

```a[t_]=D[V[t],t]

-(y0 Cos[t])```

10 ) Plot the displacement, the velocity and the acceleration

```ploty=Plot[y[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[1,0,0]];```

Plot of Displacement verses Time: y(t)=y0 sin(t)

Plot of Velocity verses Time: v(t)= - y0 sin(t)

```plotV=Plot[V[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[0,0,1]];```

Plot of Acceleration verses Time: a(t)= - y0 cos(t)

```plota=Plot[a[t]/.y0->.1,{t,0,6},
PlotStyle->RGBColor[0,1,0]];```

11 ) Combine the plots

`plotall=Show[{ploty,plotV,plota},PlotRange->{-.1,.1}];`

y(t)=y0 cos(t)

v(t)= - y0 sin(t)

a(t)= - y0 cos(t)